A matrix model for simple Hurwitz numbers, and topological recursion

TL;DR

This paper presents a new matrix model for simple Hurwitz numbers, computes its spectral curve, and proves a conjecture linking Hurwitz numbers to spectral invariants of the Lambert curve, advancing the understanding of their mathematical structure.

Contribution

Introduces a novel matrix model for generating simple Hurwitz numbers and establishes a connection to spectral invariants of the Lambert curve, confirming a conjecture.

Findings

Derived the spectral curve of the matrix model

Calculated symplectic invariants for the model

Proved the Bouchard-Marino conjecture

Abstract

We introduce a new matrix model representation for the generating function of simple Hurwitz numbers. We calculate the spectral curve of the model and the associated symplectic invariants developed in [Eynard-Orantin]. As an application, we prove the conjecture proposed by Bouchard and Marino, relating Hurwitz numbers to the spectral invariants of the Lambert curve exp(x)=y exp(-y).